What makes an excellent teacher of mathematics?

Contributed by:
Sharp Tutor
We will be focusing on the following points:
1. The importance of effective teaching
2. Teaching effectiveness and student achievement in Australia
3. Measuring effective teaching
1. Learning from the best:
what makes an excellent teacher
of mathematics?
By Dr Bronwyn Hinz, Dr Lyndon Walker, and Mr Michael Witter
2. About Pivot
Pivot Professional Learning is an organization that aims to transform
teaching practice through a suite of data-rich tools including our
flagship Student Perception Survey. Pivot’s Student Perception
Survey asks students to provide feedback on teaching practice
at the individual classroom level, which informs teaching at the
individual, department, school, network and system level. Results
are confidential, available the day after surveys close, and broken
down in a meaningful way for each level of user. This makes it easy
to identify and respond to the changing needs of each class, year
level, department or school.
Pivot’s student survey data helps teachers, school leaders and
executives to pinpoint strengths for sharing, and growth priorities
for development.
Table of Contents
Executive summary page..................................................................................1
Background and literature review..................................................................7
Results and discussion........................................................................................9
Policy implications............................................................................................ 18
Examples of effective PCK from the research.......................................... 20
Priorities for professional learning.............................................................. 20
Conclusion........................................................................................................... 21
References cited................................................................................................. 22
Appendix A. Research Instruments ........................................................... 25
Appendix B. Statistical compendium......................................................... 25
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3. Learning from the best:
what makes an excellent teacher of mathematics?1
By Dr Bronwyn Hinz, Dr Lyndon Walker, and Mr Michael Witter2
Executive summary
Australia has many excellent mathematics teachers. However, it also faces many challenges on a
national scale: declining mathematics results; limited engagement among Australian students; a
severe shortage of qualified mathematics teachers; low levels of numeracy and a lack of confidence
in mathematics across the general population (O’Connor and Thomas 2019; IAEEA 2015; FYA and
AlphaBeta 2017; Thomson et al. 2017).
In order to enhance mathematics instruction, engagement and outcomes across Australia, it is
imperative we learn from our top mathematics teachers, as well as invest in high-quality pre-service
teacher education and the provision of ongoing and relevant professional learning throughout a
teacher’s career. But what are the key indicators and practices Australia should seek to spread?
Pivot Professional Learning (Pivot), in consultation with the Australian Association of Mathematics
Teachers (AAMT), undertook an exploratory study of the characteristics of Australian mathematics
teachers to better understand what helps and hinders great teaching so that we can learn from the best.
Using a short, opt-in questionnaire, we collected information on factors the literacture suggest could
make the biggest contribution to quality teaching - qualifications, experience, professional learning,
teaching beliefs, self-efficacy and collective efficacy. 986 teacher from around Australia particpated in
this questionnaire. Results data from the Pivot Student Survey3 for each teacher, where available, was
then accessed and linked to the questionnaire (n=673), with teachers’ consent, to explore relationships
between items. Those teachers with the highest aggregate scores from the Pivot survey we identified
as ‘Top Teachers’. Follow-up qualitative interviews using a set of semi-structured questions were
then held with a small but representative sample of Top Teachers. These interviews were to gain
greater insights into features of their training, professional learning, pedagogical beliefs and teaching
practices that appear to make the biggest difference in supporting students to learn mathematics.
1 This paper was presented at the AAMT Conference, Brisbane, 9 July 2019, in session titled ‘Great, confident maths teaching: what are
the indicators, training and PL supports and practices?’
2 Dr Bronwyn Hinz is Director of Research and Development at Pivot Professional Learning, and Honorary Fellow at the Melbourne
Graduate School of Education. Dr Lyndon Walker is a former academic, OLT Citation holder, and statistical consultant for Veritate Data
Science. Michael Witter is Director of National Curriculum at Teach For Australia and a PhD candidate with Prof. John Hattie focusing on
dimensions of teacher quality. The authors express their deep appreciation to all 986 study participants, and also to the AAMT and seven
external reviewers (four of whom were expert math teachers) who each provided useful feedback that contributed to the accuracy,
clarity and overall quality of this paper. The authors’ take responsibility for any remaining faults.
3 Pivot’s evidence-based, validated tool is an established method for measuring student perceptions of teacher effectiveness against
each of the five Australian Professional Standards for teachers developed by the Australian Institute of Teaching and School Leadership
(AITSL Standards). It consists of a five-minute, 25-item survey completed by school students on their perception of the teaching practices
they experience in each classroom, with confidential, next-day reports available for each teacher, and aggregated results reports
available for school leaders. (Available as Appendix A2)
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4. Key Insights
Although this was an exploratory study, the large number of participants and statistically significant
findings, substantiated with qualitative interviews, provide insights into the attributes and training of
Top Teachers of mathematics. It is our hope to explore these more deeply in follow-up studies.
Top Teachers • Have strong connections with their students, characterised by
mutual respect and high expectations. They understand where each
student is at, how to support them, engage them and stretch them.
This student-teacher connection appears to be a higher-order factor for
great mathematics teaching. This supports the idea that students filter
their learning experiences and perceptions of quality teaching in part
through the quality of student-teacher interactions (Hamre et al 2014;
Kufheld 2017).
• Emphasise the importance for mathematics lessons to include
collaborative work (where students can build and reinforce their
understanding from the understanding of their peers), “hands-on”
problem-solving, and linking the learning to real life in order
to trigger and sustain student engagement. Top Teachers use their
strong connections with students to design and adapt their lessons to
effectively include these elements.
• Recognise that deep learning (exploring concepts) and surface
learning (recalling facts) are both important and connected in
building mathematical knowledge and skills.
Professional learning • Newer teachers had lower results for classroom management
indicators, which can make teaching and learning more difficult.
and teaching While this is true, there were novice and experienced Top Teachers (as
measured by Pivot’s Student Perception Survey) which suggests that
experience both more-experienced and less experienced teachers can learn from
each other.
• Teachers that had participated in maths-specific professional
learning were more confident than teachers that had not done so
when it came to:
• Demonstrating, modelling and explaining skills and concepts to
• Effectively using a variety of assessment and feedback strategies
• Accessing the resources they need.
• Mathematics teachers see professional learning as most valuable
when they can identify its relevance for their students and classes
they teach.
• Teachers reported the best types of professional learning were
ongoing and promoted both action and reflection, both as an
individual teacher, and with colleagues, to support and fine-tune
implementation of PL strategies in their classes.
• Teachers who were members of professional associations - such as
the Australian Association of Mathematics Teachers - were more likely
than non-members to agree that they can access the resources they
need to teach effectively.
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5. Qualifications and • The uniting characteristic of Top Teachers was their ability to
form strong connections with students, which was the foundation
other characteristics for creating engaging lessons, determining the extent to which each
student understood what was taught, and identifying what supports
or extension are required to better enable each student to master the
mathematical concepts and skills. This was a stronger indicator than
years of teaching experience.

scores of male and female teachers.
• Teachers with higher degrees in mathematics reported greater
in demonstrating, modelling and explaining mathematical
concepts and skills, and were more likely to agree that their teaching
helped students to apply information in new scenarios. However, this
and connects to the point below.

mutually exclusive. Excellent mathematics teachers need both sets
of expertise in order to combine deep conceptual knowledge with the
ways, and to provide a myriad of opportunities for students to practise
and apply mathematics in a way that triggers and sustained their
This study reinforced that Australia could do better at sharing and building excellent mathematics teaching
within schools, between schools, and across systems. The study also reinforced that mathematics expertise
skills and that excellent teaching requires both. This has implications for both pre-service teacher education and
ongoing professional development and learning.
Figure 1
teachers combine maths
expertise with teaching
expertise enabled through (PK) KNOWLEDGE (CK)
strong connections with (PCK)
their students ‘The how’ ‘The what’
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6. Professional • Create and curate an online space and repository for mathematics
teachers to share and discuss teaching resources, strategies, and
associations and professional learning from Top Teachers. While this could include
contributions by relevant academics and other mathematics
school systems can: professionals, the content would be mostly by teachers, for teachers.
This has the potential to increase teacher use and engagement, as they
can more readily see relevance when it comes from fellow teachers.
• Develop professional learning opportunities and programs that
support teachers to build strong, interactive connections with their
students in their mathematics classes.
• Continue to promote and support professional learning that supports
the cultivation of collaboration, creativity and critical thinking as they
apply in mathematics.
• Recognise that quality teaching integrates surface knowledge (of facts)
and deep learning (exploring concepts).
Schools can: • Support early career teachers to build their classroom management
skills, as this is a critical factor in being able to teach well. This could
include more intensive scaffolding and mentoring of new teachers
for the first five years, and more team-teaching (where teachers
teach in pairs or small groups), which provides opportunities for
more experienced teachers to share their expertise in classroom
management and student engagement, and newer teachers to share
research and strategies learnt in their pre-service education and
teaching rounds.
• Encourage all mathematics teachers to participate in both formal and
informal collegial observations (with clear focus, debrief and follow-up
plan and actions).
• Support teachers to collaborate in their use and development of
teaching resources and strategies, as well as to share relevant research
they’ve read, as part of their ongoing professional learning.
• Identify and celebrate existing Top Teaching teams as established by
those teachers’ students (for example using Pivot survey data and other
achievement data) to reveal the most effective practices for teaching
different concepts, different year levels and students of differing ability
• Recognise that highly experienced teachers and new teachers can both
be excellent teachers of mathematics, and encourage all teachers and
teams to share strategies for articulating key concepts and engaging
students in maths.
• Connect their mathematics department with that of other schools in a
mathematics Community of Practice, to share strategies, resources and
to build collective efficacy.
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7. Governments, • Work alongside schools to provide additional support, resourcing and
development opportunities for early career teachers, including team
school systems and teaching where this is desired by the school.
partner bodies can: • Support out-of-field mathematics teachers to upskill, and other
teachers wanting to consolidate or extend their knowledge, through
supporting their access to specialised, bridging courses.
• Continue to promote and support - through policies, curriculums and
resourcing - the importance of student voice, and of collaboration,
creativity and critical thinking in mathematics and related subjects.
• Curriculum authorities could consider more emphasis on
multidisciplinary approaches to learning mathematics, and guidance
to support this, as a way of encouraging engaging and contextualised
mathematics learning.
• Consider reducing the amount of content in senior curriculums,
to allow for deeper exploration and mastery of key concepts, and
continuation of best practice teaching and individualised approaches
to support all students to achieve the best.
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8. Australia has many excellent mathematics teachers. But a variety of factors contribute
to a situation where not every student or teacher can reach their full potential in
learning or teaching this vital subject.
Pivot Professional Learning, in consultation with the Australian Association of Mathematics Teachers, decided
to learn from the best in order to build and share the expertise of our top teachers, and build the mathematical
skills and knowledge of our students. This paper presents the findings from our short, exploratory study seeking
to identify characteristics of top teachers (qualifications, beliefs, practices, professional learning and more) and
relationships between these and the learning experiences of their students. This was a mixed-methods study
involving quantitative and qualitative components. Research instruments included a questionnaire of teachers,
with items drawn from a review of the literature, results from a student perception survey on effective teaching
practices, statistical analyses and qualitative interviews.
Results indicated that teachers that had participated in maths-specific professional learning, or held higher
degrees in mathematics were more confident across a range of measures; that strong and positive student-
teacher relationships are a foundation for highly effective teaching and learning, and that Top Teachers
needed both deep conceptual knowledge and teaching expertise. These findings reinforce the importance of
Pedagogical Content Knowledge, which enables teachers to provide interactive lessons with opportunities for
collaboration, real-world problem solving and creativity, to build students’ engagement, confidence, skills and
knowledge in mathematics. This has implications for teachers, universities, schools, governments and other
education stakeholders.
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9. Background and literature review
The importance of effective teaching
Teaching effectiveness is the single most important in-school factor influencing student engagement and
achievement (Hattie 2013). A large proportion of the variance that occurs in student achievement is attributable
to their teachers, and teachers are probably the greatest school-related factor influencing student achievement
gains (Sanders and Horn 1998, Darling-Hammond 2000, Rivkin, Hanushek and Kane 2005). The magnitude of
the effect of the teacher on student achievement can be as much a full year’s difference (Hanushek 1992). Rowe
(2003) provided a strong evidence base demonstrating the importance of teacher effectiveness in influencing
student outcomes in the Australian context, as did Leigh (2010) who found that teachers in the 75th percentile
were achieving in three quarters of a year what a 25th percentile teacher achieved in a full year, while teachers
in the 90th percentile achieved in a half-year what 10th percentile teachers achieved in a full year.
Teaching effectiveness and student achievement in Australia
Initiatives and investments in teacher effectiveness have featured strongly in state and Commonwealth
education policies. Among other things, this has resulted in the creation of the Australian Institute of Teaching
and School Leadership (AITSL) and its professional standards for teaching, as well as near-continuous reforms
over the past decades to strengthen pre-service teacher education. Unfortunately, there is no clear evidence
these reforms have improved academic outcomes for students in general, and specifically in mathematics. The
2015 PISA results indicated that Australian students’ scores continue to decline from prior years, continuing
a downward trend that has been occurring now for over a decade. Meanwhile data from TIMSS (Trends in
International Mathematics and Science Study) show Australian students flatlining while other countries improve
(Thomson, Wernert et al. 2016). Further to this, only slightly more than half of Australian students are achieving
the National Proficient Standard in mathematical literacy (Thomson, De Bortoli et al. 2017).
Many factors contribute to these lackluster results. These factors include: poor policy coordination between
state and Commonwealth levels of government; inadequate or poorly targeted funding (Hinz 2016); a severe
shortage of qualified mathematics teachers (O’Connor and Thomas 2019); teacher and school leader workloads
(OECD 2019); and poor or incomplete implementation at school and classroom levels, and miss-match between
the program and the problem or context (Evidence for Learning 2019). An absence of evaluations or research
exploring which of these factors most affected the policies seeking to improve teaching and student learning
means we are left to speculate as to what reforms, programs and actions at all levels could make the biggest
difference to teacher effectiveness and student learning growth in mathematics.
Measuring effective teaching
Teaching is a complex profession involving a number of interrelated skills and attributes and influenced by
a number of contexts and other variables. We, the authors, take a complementary approach to measuring
and understanding teacher effectiveness, reflecting this complexity and ongoing debate. In doing so, we
follow other Australian and international researchers (Darling-Hammond 2000; Rice 2003, Wayne and Youngs
2003; Ingvarson and Rowe 2008; Naylor and Sayed 2014) in drawing upon Strong (2011), whose review of the
research on this question identified four basic characterisations or conceptions of quality: teacher qualifications
(including their degree, program and experience), personal attributes (such as kindliness, flexibility, patience),
pedagogical skills and practices, and teacher effectiveness (the value added to student academic achievement).
Our exploratory study investigated which of these characteristics and indicators are most useful in
understanding and supporting excellent mathematics teaching in Australia, with particular reference to
experience (years teaching overall, and years teaching maths), highest relevant qualification, professional
learning, attributes and beliefs (including self-efficacy and collective efficacy, and surface versus deep learning).
We also explored the connections of these indicators with the learning experiences of their students as an
indicator of quality teaching practices in these teachers’ classrooms.
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10. In doing so we drew upon the landmark MET study (2013) and more recent Wallace et al. study (2016) which
found that carefully designed and statistically validated student perception surveys on specific teaching
practices were found to be a reliable indicator of effective teaching practices and their impact on student
performance. This finding has further reinforced by Australian research indicating that student feedback is
among the “most powerful” kids of feedback for effective teaching (Hattie 2008); that successful systems of
feedback including from students, can increase teacher effectiveness by as much as 30 per cent (Jensen 2011),
and OECD research which found that student feedback surveys are a key element of teacher professional
development systems in high performing countries such as Finland and Singapore (2013).
Invitations to complete a short, online questionnaire on their mathematics teaching experience, training,
and beliefs were sent by email to all teachers on Pivot Professional Learning’s database who had taught a
mathematics class in an Australian school in 2017, 2018 and 2019 (n=2683). These invitations were sent in
March and April 2019. Information on the study with an invitation to participate was also distributed by the
AAMT to their members.
This teacher questionnaire covered participants’ teaching experience, qualifications, professional learning,
teaching and learning beliefs, self-efficacy and collective efficacy (Teacher questionnaire items are attached
in Appendix A1). This questionnaire was designed by the authors, in consultation with the AAMT, to capture
key items identified from the literature as having a potential bearing on teaching quality. A total of 986
mathematics teachers participated in this questionnaire, and came from all Australian states, territories and
school sectors.
Results data from the Pivot student survey for each teacher, where available, were then accessed and linked to
the questionnaire data (n=673), with teachers’ consent, to explore relationships between items. (Student survey
items are attached as Appendix A2). While this study did not link this data to student performance data, we
are confident the student survey instrument is a reliable indicator of effective teaching practies and of quality
and effective teaching practices, due to extensive validity testing across Australia, sophisticated design of the
instrument meeting all key criteria for such surveys, and their impact on student on student performance and
engagement across a diversity of school contexts.
Descriptive statistics are presented in the form of means and percentages, with t-tests, chi-squared tests,
analysis of variance (ANOVA), and multivariate multiple regression methods used to examine the bivariate and
multivariate relationships between teacher demographics, teacher perceptions from the questionnaire, and
student perceptions from the Pivot instrument. A combination of confirmatory and exploratory factor analysis
was used to determine whether the various survey items could be grouped into broader categories.
Follow-up interviews were then conducted in June with a representative selection mathematics teachers with
the highest aggregate Pivot student survey scores (i.e. the most effective as rated by their students based on
teaching practices they experienced) to explore in greater depth the relationships found in the quantitative
analysis. The 20 highest-scoring mathematics teachers who had also indicated interest in a follow-up interview
were emailed invitations to be interviewed. Four teachers participated in semi-structured telephone interviews
of 30-45 minutes. (Interview questions are attached in Appendix A3.) Of these, two were male, two were female,
they had taught for 2 years, 20 years, 43 years, and 45 years, in government and non-government schools, and
in metro and regional locations. This diversity reflects the pool of the 20 Top Teachers.
Participation in all elements of this study were voluntary, and participants were informed with a plain language
statement, questionnaire preface, as well as in opening statement in research interviews that their participation
was voluntary, that their data was confidential, and that they could withdraw at any time without prejudice.
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11. Results and discussion
Study participants: well educated, but not necessarily in mathematics, or
members of professional associations for mathematics teachers
The figures below show the descriptive statistics for the sample in the teacher questionnaire. The sample was 58
per cent female, with an average of 15.6 years teaching and an average of 14 years teaching mathematics. These
distributions were right skewed, with a number of very long serving teachers (maximum 55 years). The median
number of years teaching was 13, with a median of 10 years teaching mathematics.
Over 99 per cent of research participants had at least a Bachelor degree, with nearly 70 per cent holding more
than a Bachelor degree: e.g. Bachelor plus a Graduate Certificate in applied field, Masters, or a PhD or Doctorate.
Only 54 per cent of participants majored in mathematics or held a mathematics qualification, indicating that a
considerable proportion would be considered an “out of field” teacher, consistent with the findings of O’Connor
and Thomas (2019) and TIMSS data (2015) which found 22 per cent of Australian Year 8 students were taught by
out-of-field teachers, much higher than the international average of 13 per cent.
Most teachers in the sample (88 per cent) had completed some kind of maths-specific professional learning or
development programs, but only 30.5 per cent were a member of a professional body.
3.3% | n=33
Masters Bachelor
21.0% | n=207 30.5% | n=301 NO 45.9% | n=453
qualification or major
YES 54.1% | n=533
NO 12.3% | n=121
Participated in
maths-specific YES 87.7% | n=865
Bachelor plus professional learning
44.8% | n=442
58.0% | n=572
Member of AAMT or similar NO 69.5% | n=685
GENDER professional body for
teachers of mathematics
Male Other YES 30.5% | n=301
41.8% | n=412 0.2% | n=2
N Missing Mean SD Min Q1 Median Q3 Max
Years Teaching 972 14 15.63 11.76 1 6 13 25 55
Years Teaching
965 21 13.97 11.39 1 5 10 21 55
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12. Overall, mathematics teachers in this study were confident
Overall the mathematics teachers in the study were very positive, with most nominating that they mostly agree/
strongly agree with the statements below:
• I can get my students to follow the classroom or school rules
• I am able to effectively use a variety of assessment and feedback strategies
• I am able to access a wide variety of classroom resources that support my teaching content
and pedagogy
• I am confident when it comes to demonstrating, modeling and explaining mathematical
skills and concepts to my students
Most participants also indicated positive self-efficacy around their teaching, and agreement on the importance
of deep learning. Participants were slightly less positive on the questionnaire items about “teachers in my
school” compared to themselves, with more opting for “slightly agree” or “mostly agree” rather than “strongly
agree” in their answers to questionnaire items. This sample could be biased due to the opt-in nature of the
questionnaire, with more confident teachers being more inclined to participate. Table 1 in the Appendix B
shows the responses to the teacher questionnaire items about self-efficacy, the teaching environment, and
views on teaching.
Teacher confidence bolstered by relevant qualifications and professional learning
Teachers who had participated in maths-specific professional
learning or development were nearly twice as likely to strongly
agree they could access a wide variety of classroom resources that
support [their] teaching content and pedagogy.
The teachers that had participated in maths-specific PL were also more confident that they were “able to
effectively use a variety of assessment and feedback strategies”, “confident when it comes to demonstrating,
modeling and explaining mathematical skills and concepts to my students”, and “get my students to follow the
classroom or school rules”.
Teaching confidence was also positively related to having a qualification in mathematics with 64 per cent of
maths-qualified teachers strongly agreeing that “I am confident when it comes to demonstrating, modeling and
explaining mathematical skills and concepts to my students” compared to 39.7 per cent of teachers who would
be considered to be teaching “out of field”. Those teachers who had a mathematics qualification were also
more likely to agree that “It is important to teach (and for learners to know) the material that will be covered in
Indicators such as years of teaching, gender, and highest qualification did not have a significant relationship
with the self-efficacy measures in the teacher questionnaire. This analysis used chi-squared tests of
independence to measure the statistical significance of the relationships between items in the teacher
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13. Student perception survey results reveal student-teacher connection as a
mega-factor of effective teaching
A validation of the Pivot results for the mathematics teachers in the survey shows that a student-teacher
connection appears to be a higher-order factor for student learning and engagement, and also teacher
engagement. It is speculated that students’ perceptions of their teachers’ effectiveness are heavily influenced
and filtered by the quality of student-teacher interactions, a phenomenon that has been identified in other
research ((Hamre et al 2014; Kufheld 2017; Wallace et al 2016)). A critical implication when considering students’
experience in the classroom and of their teachers is that when the quality of those interactions are weaker, this
is likely to affect students’ perceptions of most if not all of their teachers’ individual practices.
Table 2 in the Appendix B shows the Pivot scores for the 673 survey participants who had received student
feedback from maths classes in 2019. The average student feedback scores for mathematics teachers were
generally similar to the population norms, with the largest differences seen in the items “This class keeps my
attention - I don’t get bored”, “This teacher makes what we are learning interesting”, and “In this class, the
teacher helps me build my vocabulary” - each of which had a lower average with a notable, albeit small, effect
size (Cohen’s d of 0.2 to 0.25), than the population. The latter result could possibly being a product of the
content taught in secondary school maths curriculums, while the former may be related to the views of an
average student on their maths classes relative to other classes. (Compulsory subjects tend to receive lower
scores than elective subjects, reflecting perhaps research that greater student voice and choice in what and
how they learn has positive impact on engagement and learning outcomes (Quaglia 2016; Hattie 2009). The
rank of the questions were also generally similar between the mathematics teacher sample and the population
Follow-up qualitative interviews shed light on the connection between robust teacher-student connections
and maintaining student engagement and learning growth, as well as some of the risks when teaching
Maths as a subject is one where you can fall into a trap of
being very procedurally-focused; in which students gain
the view that it’s very black and white and has no room for Asked what strong teacher-
creativity. And some teachers are able to teach effectively this student connection look like,
way. However, maths teachers need a very deep conceptual the Top Teachers elaborated:
understanding of the concepts, so that the student’s
procedural understanding can be enriched by the conceptual.
A deep conceptual understanding also enables maths Very open, honest dialogue,
teachers to be flexible and welcoming of student-led supportive ...A really positive
discussions or suggestions of alternative procedures. relationship is key to all of
“That’s interesting, let’s see why that worked”, instead of “no this, so kids believe that you
that’s not the way I showed you”. are there for them to get them
Participant #1512337
where they want and need
to be. Moving around and
[Maths teachers] need a very good understanding of the questioning them. A super-
maths themselves, and ability to articulate it clearly for interactive environment.
each student, and understanding of a myriad of different Participant #1605142
ways that kids can learn it, and how to hook it in to kids
thinking. This is because so many kids find it difficult, and
it’s vital to develop their confidence and engagement. This all
comes from a good relationship, knowing your kids, and them
knowing you know them, care for them and will support them.
Participant #1605142
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14. Less-experienced teachers tend to encouter more challenges with classroom
management and engagement
Figure 2 shows the student perceptions that varied most by the number of years of mathematics teaching
experience. The graphs show the mean scores from the items, with the error bars showing the standard error for
that group. Teachers who had taught mathematics for less than two years scored lower on student survey
items relating to keeping a student’s attention, explaining difficult things clearly, student behaviour, and
making learning interesting. A follow up one-way ANOVA showed that the mean difference between teachers
with under two years of experience and the other groups was statistically significant for keep student attention,
explaining difficult things, and students’ in-class behaviour. Students rated teachers with between five and
10 years most highly on the first three, and the most experienced teachers had students reporting the best-
behaved classrooms.
Figure 2
Student perceptions of effective teaching, by years of teacher experience teaching mathematics
Q3 This class keeps my attention - Q10 This teacher makes what we are
I don’t get bored learning interesting
3.4 365
(0-2] (2-5] (5-10] (10-20] (20-55] (0-2] (2-5] (5-10] (10-20] (20-55]
Years teaching Mathematics Years teaching Mathematics
Q7 This teacher explains difficult things clearly Q17 In this class the students are well behaved
3.95 3.8
3.80 3.4
3.75 3.3
(0-2] (2-5] (5-10] (10-20] (20-55] (0-2] (2-5] (5-10] (10-20] (20-55]
Years teaching Mathematics Years teaching Mathematics
It is noteworthy that one of the items listed here relates specifically to student and classroom behaviour.
Managing classroom behaviour of students is one of the recurrent themes highlighted in multiple major studies
of graduating and early career teachers. The SETE Project found that graduate teachers felt underprepared in
classroom management compared to nearly all other elements of their teaching (Mayer et al. 2015). The most
recent OECD report revealed that just 45 per cent of the thousands of Australian teachers surveyed reported
that they felt prepared to manage their classrooms (2019). The data suggests a need for further improvement
and emphasis in pre-service teacher education as well as early career teacher support, including teachers
of maths, in more effective classroom management strategies. This hypothesis was confirmed in follow-up
interviews with teachers.
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15. A Top Teacher with 43 years experience emphasised that mathematics teachers need to be “100 per cent
confident” in managing behaviour if they want to support kids to learn, saying:
If you can’t manage behaviour, if the class isn’t a positive learning environment, the kids won’t be able to
engage and learn and make progress.
Participant 1605143
Another Top Teacher with 20 years experience teaching, plus experience as a head of a mathematics
department in a school noted that she wasn’t ready for the classroom management challenges she
encountered in schools.
When you’re an early career teacher, you’re pretty much thrown in the deep end very quickly by yourself in
mainstream schools. I’d like to see greater investment in new teachers having opportunity to team teach
for a while, not be the only teacher in the classroom, and not just for the first five weeks with mentor.
Participant #1530830
These findings are broadly consistent with other Australian studies on teaching, which indicate that teaching
quality does seem to improve with experience but only for the first five or so years in the classroom (e.g. Rice
2003), and that after that point, experience accounted for less than one per cent of the variation in teaching
quality (Leigh 2010). It also aligns with international research suggesting that the effects of experience are at
best small and at worst statistically insignificant (Goldhaber and Anthony 2007; Goldhaber and Brewer 2000).
It reinforces that more can and should be done to support teachers in their first five years, especially when
it comes to creating and sustaining productive learning environments in their classrooms, a prerequisite to
deeply engaging with mathematics.
Higher qualifications of teachers can have an inverse (negative) correlation
Figure 3 shows the Pivot Student Perception Survey items that varied most by highest qualification. Relative to
Bachelor-qualified teachers, those with Bachelor plus a Graduate Diploma and Masters-qualified teacher groups
scored lower, on average, on the items relating to student understanding and interest. The pattern doesn’t
extend to the PhD group, but it should be noted that the PhD group is relatively small (which also accounts for
the larger standard errors).
Figure 3
Student perceptions by teacher highest qualification
Q2 This teacher cares about the students’ Q10 This teacher makes what we are
point of view learning interesting
4.10 3.8
3.95 3.6
Bachelor Bachelor + Masters PhD Bachelor Bachelor + Masters PhD
Highest Qualification Highest Qualification
w w w.pivotpl.com 13
16. Q7 This teacher explains difficult things clearly Q5 This teacher knows when the class
understands and when we do not
3.9 3.95
3.8 3.85
Bachelor Bachelor + Masters PhD Bachelor Bachelor + Masters PhD
Highest Qualification Highest Qualification
It has been found that experts can lose their sensitivity to how difficult a task can be for novice learners, which
can increase the difficulty they experience in teaching skills and concepts to their students, a phenomenon
known as “the curse of knowledge” (Hattie and Yates 2013). Given that the Pivot survey items are consistent with
teacher sensitivity to learner needs and interests, it may be the case that some teachers with expert knowledge
of maths find it challenging to fully understand and respond to the potential misconceptions and challenges
their novice learners grapple with, or were still in their early years of teaching.
Two of the Top Teachers in this study were primary school teachers who later moved into secondary school
mathematics teaching roles. These teachers undertook university-provided, government-funded graduate
diplomas in mathematics teaching, which they rated very highly and emphasised were extremely valuable
equipping them with the relevant conceptual knowledge as well as strategies and tools to communicate these
concepts to students of different ages, abilities and levels of engagement. These teachers also stressed that
the fact that the course went for a year or two, involved face-to-face time, and opportunities to deeply reflect,
apply, refine and collaborate with peers in the course, contributed to this training being so very useful.
[the Graduate Diploma in teaching maths to students in years 7-10] provided me with great data and
resources and strategies. Most definitely the best PL for maths I’ve done. Weekend intensives. Made
connections with other teachers from all over Victoria.
Participant #1512446
Research on the impact of teacher certification or registration on effective teaching has been inconclusive, with
some studies finding strong positive effects (Darling-Hammond 2000), and others small, mixed or negative
effects (Rice 2003; Wayne and Youngs 2003). Wayne and Youngs’ review noted that mathematics teachers,
unlike those in other subjects where results are unclear, do produce better results for their students across these
types of measures of subject area knowledge and experience, and our results suggest that specialised training
in mathematics instruction appears to be the key factor (as opposed to teaching more generally, or in other
This study did not find significant relationships between a teacher’s gender, participation in professional
learning programs, or possessing a mathematics qualification, and their student survey scores on different
Pivot items. This gender finding is strikingly at odds with findings of studies on student perception surveys
completed at universities of their lecturers and tutors which indicate discrimination against women (Mitchell
and Martin, 2018; Fan et al. 2019). It is also borne out by analysis of the Pivot data set more broadly (including
teachers of other subjects) which does not indicate that school students score teachers differently based on
their gender.
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17. The results on professional learning could be explained by the fact that not all professional learning programs
are relevant to the needs of those teachers or of their students, or that the PL is not reinforced with time
for teachers to apply and refine the learning and strategies taught, or time to discuss with colleagues. It is
hypothesised that where these features are present, the PL is highly valuable and supports quality teaching,
but where PL is a “one-off” session with minimal application, minimal collaboration and practice opportunities,
and no follow-up, it had negligible or negative influence on teaching and learning. These hypotheses were
confirmed in qualitative interviews:
I don’t really value “sit and listen” PL. The most valuable PLs [Some teachers need] more
are things you can easily link in to the classroom and support in conveying maths
put into practice, such as lesson observation, discussing concepts to students. And others
feedback with a colleague, then implementing in class. also need conceptual training in
Participant #1512337 maths. PL needs to be tailored
in the same way that teachers
tailor lessons and activities for
The best PL I ever did had time after the PL to follow-up, their students, scaffolding them.
to explore how to embed, to check-in with course teachers. Participant #1512337
Participant #1530830
Teacher versus student perceptions
The final component of quantitative analysis was an examination of the relationships between teacher
perceptions from the teacher questionnaire and student perceptions as measured by the Pivot instrument, with
particular focus on items that measured similar concepts.
The pattern that emerged from the data was there seem to be two distinct cohorts of teachers who tend to
receive higher student perception ratings - the teachers who have the highest self-efficacy scores (rating
themselves with ‘strongly agree’) and a smaller group who were rating themselves with ‘strongly disagree’ on
the efficacy questions. This could suggest that some mathematics teachers suffer from “Imposter Syndrome”
where they don’t believe they are doing a good job (or at least are not confident) despite the higher ratings
from their students. It could also suggest that some students are overstating effectiveness. In other words, while
student perception is a reliable indicator of effective practice, at an individual or small sale it is not infallible),
which is why Pivot recommends whole class provides feedback, rather than a hand-picked handful of students.
This reinforces the need for clear, strong data drawn from multiple measurement instruments - including Pivot’s
Student Perception Survey, for a fulsome understanding of effective teaching.
Figure 4 plots the mean scores for This teacher is knowledgeable about the topics in this subject, with standard
error bars, against the survey item I am confident when it comes to demonstrating, modeling and explaining
mathematical skills and concepts to my students (1=Strongly Disagree, 6=Strongly Agree). The highest
performing groups are the teachers who answered in the extreme, while the worst performing group are the
“slightly disagree” cohort (corresponding with 3 in Figures 4 and 5).
Figure 4
Teacher 4.4
student preception
confidence 4.3
vs Student 4.2
of Teacher 4.1
Knowledge 4.0
1 2 3 4 5 6
teacher confidence Demonstrating, modelling and explaining
w w w.pivotpl.com 15
18. Figure 5 plots the classroom control items, In this class, the students are well behaved (1=Strongly Disagree,
6=Strongly Agree), from the Pivot student results, against the teacher survey item, I can get my students to
follow the classroom or school rules. Again, the “slightly disagree” cohort has the lowest average scores whilst the
teachers at each extreme have the highest.
Figure 5 4.0
student preception
and Teacher
Perceptions 3.5
of Student
1 2 3 4 5 6
Teachers agreeing they can get their students “to follow the classroom
or school rules”
Pivot student survey results are discussed further in implications, but the project was able to confirm the factor
structure of the five professional standards Pivot groups its student perception survey questions within with
high estimates of reliability (Standard One alpha = .95, Standard Two alpha = .94, Standard Three alpha = .95,
Standard Four alpha = .90, Standard Five alpha = .95.).
The noteworthy findings here are that students view classroom management as more distinct than any other
element of their teachers’ practices, and that strong, positive teacher-student connections are a mega-factor of
effective teaching and learning.
Teaching beliefs
Items on self-efficacy and collective efficacy, as well as on learning, were gathered and explored in our study,
to probe findings of earlier studies that found that teachers’ beliefs significantly influence their planning
instructional decisions and classroom practices, including behaviour management (Kagan 1992, Pajares 1992,
Calderhead 1996, Hattie 2003).
The items in the teacher questionnaire that had been hypothetically labelled as self-efficacy and collective
teacher efficacy grouped together with no obvious sub-factors. These items are in Appendix A1. Estimates of
reliability were sufficiently high to have confidence in the total scores from the scales (self-efficacy alpha = .88,
collective teacher efficacy alpha = .87). This is consistent with earlier research finding more effective teachers
tend to have higher self-efficacy (Woolfolk, Rosoff and Hoy 1990; and Henson 2001, Ross 1994, Ross 1998, Ashton
and Webb, 1986, Muijs and Reynolds 2001).
Beliefs about teaching and learning were originally segmented based on deep versus surface, with teachers
asked how much they agreed with the four statements below:
• Learning is building up knowledge by acquiring facts, information and skills
• It is important to teach (and for learners to know) the material that will be covered in
• Learning is about seeing things in a more meaningful way.
• My teaching helps students to apply information to new situations or problems
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19. Factor analysis grouped all items together into just one factor that united surface and deep beliefs about
teaching and learning, with high estimates of reliability (alpha = .83). This finding indicates that in general,
teachers who strongly endorse surface views of teaching and learning are also much more likely to
endorse deep views, as opposed to viewing these as mutually exclusive. Between 85 per cent and 90 per
cent of participants agreed that both deep and surface components of teaching and learning were important.
This result aligns with earlier studies finding an empirical link between improved student outcomes teachers
that follow both deep and surface focused teaching and learning beliefs (Entwistle 1997, Biggs, Kember et al.
2001, Biggs and Collis 2014).
This finding is novel, because it shows that teachers who push for deep versus surface teaching and learning
are not noticeably more effective by any measure (self-efficacy, Pivot results, etc.). Further, it is noteworthy
that mathematics teachers do not see surface learning and deep learning as dichotomous, but instead
consider surface knowledge (facts) and deep learning (concepts) as both integral to good teaching. Again,
our qualitative interviews shed light on this phenomenon, with a disjuncture between “best practice” teaching
and “exam and curriculum driven teaching”. Top Teachers described excellent mathematics teaching as
combining explicit instruction with guided inquiry, individualised tasks, choice, and collaborative
exercises to explore and master concepts and skills. But they all emphasised this was very difficult to do
in Year 11 and 12 classes, where the extensive content that needed to be covered allowed little time to delve
deeply into the concepts or their application. When it came to Years 11 and 12, there was a perception that
“good teachers” were those that managed to get through the content required for the exams:
I still get really frustrated that we have
different system Years 7-10 (where there
Good teaching and learning involves
is lots of individualised student learning
the opportunity to wonder, and
and more time to explore and practice),
to practice techniques that arise
then the regimented approach for Years
from this wonder, and balancing
11 and 12. The policy for mathematics
conceptual discovery and explicit
needs to change. I’m putting in so much
teaching of theory, connections
extra time, ploughing through everything
and practice…[but] there is a gap in
required for their exams [but it’s] sink or
between what makes good teaching
swim as there’s so much to get through. You
and the systemic requirements of
do your best to support the kids individually,
Years 11 and 12, which are in conflict.
but too much curriculum to cover to do
Participant #1530830
Participant #1605142
[what does good mathematics teaching and learning look like?] “Explicit modelling, but with a
strong element of inquiry as well [and] having students involved. Prioritise them thinking about
what is being taught, rather than giving simple answers. Ask them questions starting with “what if”
and “why”, such as, ‘What would it look like if? or ‘Why did that work? [...Teaching senior years] is less
scaffolded, and more about getting through content. I would love to spend more time with the students,
deeply exploring the curriculum content, really breaking it down and exploring it with them, but then that
would mean having to condense the next two lessons.”
Participant #1512337
w w w.pivotpl.com 17
20. Policy implications:
building and spreading expertise
Australia has many excellent teachers of mathematics, but our schools and systems and
should do better at sharing excellent teaching practices of its expert teachers within
schools, between schools, and across systems.
Results from this study indicate that while mathematics teaching expertise tends to grow with experience for
the first decade or so, and that those with decades of experience were strongest on classroom management,
some of the best teachers are new to teaching. This suggests that more and less experienced teachers can learn
from each other. All Top Teachers emphasised their desire to more closely collaborate with colleagues and in
particular to share their knowledge and skills. It was clear that they were not encouraged or provided with as
many opportunities as they felt appropriate.
When asked if they had any recommendations, all the Top Teachers interviewed independently suggested more
time and support for collaboration with other teachers (including after participating in professional learning),
and more time to prepare lessons that best met their students’ learning needs.
Teachers are really time poor. More time to prepare for classes, so teachers don’t need to take short-
cuts. And this gives more time for teachers to sit down together to discuss and try new strategies and
resources. The system should invest properly in teacher collaboration and preparation, because if
you’re slogging it out by yourself you’re in real trouble.
Participant #1530830
First step is time for collegial discussions, setting aside a time to discuss each topic or concept,
something like an hour per topic, and build up a bank of strategies for explaining and demonstrating
each concept to students.
Participant #1512337
These findings and recommendations are especially important given that this study’s results reinforce that
skill in teaching mathematics is not the same as knowing mathematics, and that the best preparation for
mathematics teaching is a university-delivered qualification that unites the “what” with the how”. This is
because while deep conceptual knowledge is important, it is also vital that pre-service teacher education
and professional learning for in-service mathematics teachers focus on how to articulate mathematical
concepts and strategies to students, especially those struggling with confidence, to build their engagement,
understanding and skills in maths. Our study reinforces the academic construct of ‘Pedagogical content
knowledge (PCK)’ . Here we defer to the elegant explanations of two leading academics who have championed
this approach.
If teachers are to be successful they would have to confront both issues (of content and pedagogy)
simultaneously, by embodying the aspects of content most germane to its teachability… It
represents the blending of content and pedagogy into an understanding of how particular topics,
problems, or issues are organized, represented, and adapted to the diverse interests and abilities of
learners, and presented for instruction.
(Shulman, 1986, p. 8-9)
w w w.pivotpl.com 18
21. It is an idea rooted in the belief that teaching requires considerably more than delivering subject content
knowledge to students, and that student learning is considerably more than absorbing information
for later accurate regurgitation. PCK is the knowledge that teachers develop over time, and through
experience, about how to teach particular content in particular ways in order to lead to enhanced student
(Loughran et al. 2012)
In our study, PCK included being able to build creativity and collaboration into mathematics lessons,
as a means to practice the application of mathematical skills and knowledge, respond to student
suggestions, including alternative ways of solving tasks. It was also described as a way of deepening
student engagement in mathematics and cultivating these critical capabilities as they apply in maths
and beyond.
Lack of deep conceptual knowledge - more commonly found in out-of-field teachers - is more likely to result
in a procedural approach to teaching mathematics. While this approach is certainly valid and useful, is best
complemented by other strategies that can do more to foster and maintain engagement, such as mathematical
challenges that prompt students to think creatively to solve “real world” problems either solo or in groups,
be it the trajectories of goal-scoring kicks that feed into elite sports coaching, or estimating water levels for
environmental actions.
The findings also indicated the vital importance of strong connections with each student, mutual respect and
continuous interactions as a critical component in quality teaching, as this enables teachers to understand
where each student is at, and how to support them. As one teacher expressed in the research interview, this is
especially important for mathematics teachers, who:
need a very good understanding of the mathematics themselves, and ability to articulate it clearly for
each student, and understanding of a myriad of different ways that kids can learn it, and hook it in to
kids thinking. This is because so many kids find it difficult, and vital to develop their confidence and
engagement. This all comes from a good relationship, knowing your kids, and them knowing you
know them, care for them and will support them.
An ongoing concern in mathematics teaching is the growing need for high quality mathematics teachers as
the number of school students skyrocket, students’ maths knowledge and skills appear to be declining in both
relative and absolute terms, and a sever shortage in qualified mathematics teachers. While this project does not
offer solutions to address future workforce needs, it does suggest that a fuller understanding of what it means
to be a quality mathematics teacher is long overdue. Although we may perceive teachers to be quite different
on the basis of their experience, their highest degree, and whether or not they are traditionally trained to teach
mathematics, students don’t tend to view these teachers as particularly different from one another. Granted
this is just one data point, but it is nonetheless an important one to consider. Yet, we know from the broader
research that maths qualified teachers and those who participate in professional learning tend to produce
better results for their students. There is a need to put this finding to the test in the Australian context to ensure
that the training and qualifications are in fact aligning with better student outcomes, whilst simultaneously
creating a space for mathematics teachers to consider the role that student voice and students’ perceptions of
teaching should have in shaping views on what constitutes excellent mathematics teachers.
w w w.pivotpl.com 19
22. Two examples of PCK and student-teacher connections in action in maths
Teachers may incorporate topics of students’ interest and relevant challenges in their design of
mathematics problems (some even incorporate the names of students.) Mathematics teachers may
also leverage grouping and paired work strategically as a way to build or reinforce connections and
collegiality between students or to leverage peer to peer support as an instructional intervention for
both lower and higher achieving students. These teachers also utilise feedback and praise, specific
to the learning in their discipline, regularly and effectively to ensure that students feel they have the
support, encouragement and recognition they need from their teacher to be successful.
Effective maths teachers utilise an array of approaches to incorporate surface learning and deep
learning to support students’ development of knowledge and skills. One common approach
is to utilise scaffolded questioning during instructions, using Bloom’s Taxonomy verbs to elicit
understanding and build conceptual knowledge from lower level to higher level questions. Another
is to structure lessons to include a combination of routine activities such as a review of previously
learnt material, including comparable problems (spiralling their curriculum), explicit instruction
of a new skill with independent practice, and a mathematical inquiry or investigative activity that
promotes more complex, critical, and creative thinking.
Priorities for professional learning
Student survey results highlight a few key implications for supporting teachers through PL. The first is the
criticality of classroom and behaviour management. Well-managed classrooms are a prerequisite to effective
instruction, and in the eyes of students’ are the most distinct teacher practice. Less-experienced teachers
tend to encounter more challenges in this area, so classroom management support tailored to the context of
mathematics classrooms for the first few years of teaching will likely improve numerous areas of teaching and
Additionally, students’ perceptions of teachers can be quite different from teachers’ perceptions of themselves
as well as the perceptions of other adults. Supporting teachers to deepen their sensitivity to students is a
major lever for improving instruction. This could involve focusing on areas such as eliciting students’ thinking
and responding to misconceptions, encouraging students to seek help and communicate when they do not
understand, and strengthening the quality of teacher-student interactions in the classroom.
The most important factor for whether professional learning achieves desired outcomes is whether teachers
participating saw it as relevant to them and their students. This was most apparent in the research interviews
and accords with other research. Professional learning should more clearly articulate the benefits and
relevance to mathematics teaching, particularly where it concerns student-teacher connections, creativity and
collaboration in maths classrooms.
The most important thing for professional learning is that teachers see it as relevant to them. There
is a tendency for maths teachers to dismiss it as something that won’t work in maths classroom, so
demonstrating relevance to teachers is critical.
Participant #1605142
The best types of professional learning identified by teachers were ongoing and promoted a continuous cycle
of action and reflection, both as an individual teacher, and with colleagues, including in the immediate period
following the delivery of the PL, to support embedding or refinement of the practices in their classrooms.
w w w.pivotpl.com 20
23. This study reinforced that Australia has many excellent teachers of mathematics,
but can do better at sharing excellent teaching practices of its expert teachers
within schools, between schools, and across systems. The study also reinforced that
mathematics expertise and teaching expertise are different but are both required for
excellence in maths teaching as these two sets of expertise work in conjunction with
each other. This has implications for both pre-service teacher education and ongoing
professional learning.
Top Teachers of mathematics combined deep knowledge of key concepts with excellence in communicating
these in engaging ways to students of different levels of understanding and interest. This excellence appears
to be supported by a bedrock of strong student-teacher connections, which this study identified as a “mega-
factor” for excellent teaching and learning. These connections are characterised by mutual respect and
continuous checking-in with students to appreciate how well they understand the material and best next steps
to sustain engagement and progress their learning.
There was a small but significant relationship between students’ perceptions of their teachers’ knowledge and
teachers’ self-reported levels of self-efficacy. While this only accounted for a small degree of the variance in
Pivot results, it suggests that higher levels of self-efficacy positively contribute to better mathematical practice
and student outcomes, a finding consistent with the large body of research on teacher self-efficacy to date.
Quantitative analysis indicated the impact of higher qualifications and professional learning showed up most
strongly on teacher confidence, but with less effect on student perceptions, which aligns with national and
international research on these indicators. This appears to be explained by the variations in quality of teaching
degrees and maths-heavy degrees, and uneven quality and relevance of professional learning for mathematics
teachers. Follow-up interviews emphasised the best preparation for teaching mathematics was maths-specific
training as part of a teaching course run by a university/teaching college, either in their pre-service teacher
education, or in a later qualification such as a Graduate Diploma in teaching mathematics, as this united the
“what” with the “how” needed for pedagogical content knowledge. For professional learning to be useful, it
had to be ongoing, directly linked to classroom practices, and include opportunities to reflect and discuss with
colleagues. This allowed the PL to be implemented, and if needed, refined, in a way the best met the learning
needs of students.
As stakeholders in education, teachers, schools, professional associations, school systems, universities and
governments all have a role to play in building and sharing mathematics teaching expertise. These are listed in
the Executive Summary, on pages four and five.
w w w.pivotpl.com 21
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students’ , Teaching and Teacher Education, 6(2): 137-148.
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27. Available upon request by emailing [email protected]
Apendix A. Research instruments:
A1. Questionnaire completed by mathematics teachers in this study
A2. Pivot student perception survey on effective teaching practices
A3. Semi-structured interview questions asked of Top Teachers
Appendix B. Statistical compendium:
B1. Teacher questionnaire items and results
B2. Pivot Student Perception Survey on Effective Teaching Practices - Scores
B3. Teacher Beliefs Factors
B4. Descriptive Statistics on teaching beliefs
B5. Relationships in the maths teacher survey
B6. Relationships between student perceptions and teacher profile descriptors
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